Eigentheory#

Definition: Eigenvector

Let \(A \in F^{n \times n}\) be a square matrix.

The eigenvectors of a square matrix \(A \in F^{n \times n}\) are the eigenvectors of the endomorphism \(f: F^n \to F^n\) whose matrix representation with respect to the standard basis of \(F^n\) is \(A\).

Definition: Eigenvalue

The eigenvalues of a square matrix \(A \in F^{n \times n}\) are the eigenvalues of the endomorphism \(f: F^n \to F^n\) whose matrix representation with respect to the standard basis of \(F^n\) is \(A\).

Definition: Eigenspace

The eigenspace of an eigenvalue of \(A\) is its eigenspace as an eigenvalue of \(f\).

Definition: Geometric Multiplicity

The geometric multiplicity of an eigenvalue of \(A\) is its geometric multiplicity as an eigenvalue of \(f\).

Definition: Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue of \(A\) is its algebraic multiplicity as an eigenvalue of \(f\).

Definition: Characteristic Polynomial

The characteristic polynomial of a square matrix \(A \in F^{n \times n}\) is the characteristic polynomial of the endomorphism \(f: F^n \to F^n\) whose matrix representation with respect to the standard basis of \(F^n\) is \(A\).

Example: \(A \in \mathbb{R}^{2\times 2}\) and \(A \in \mathbb{C}^{2 \times 2}\)

Consider the following real matrix:

\[A = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} \in \mathbb{R}^{2 \times 2}\]

Its characteristic polynomial is

\[\chi_A (\lambda) = \det \left(\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} - \lambda I_2\right) = \det \left(\begin{bmatrix}- \lambda & 1 \\ -1 & -\lambda \end{bmatrix}\right) = \lambda^2 + 1\]

This real polynomial has no roots and so \(A\) has no eigenvalues.

Now consider the following complex matrix:

\[A = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} \in \mathbb{C}^{2 \times 2}\]

Its characteristic polynomial is

\[\chi_A (\lambda) = \det \left(\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} - \lambda I_2\right) = \det \left(\begin{bmatrix}- \lambda & 1 \\ -1 & -\lambda \end{bmatrix}\right) = \lambda^2 + 1\]

This complex polynomial has the roots \(-\mathrm{i}\) and \(\mathrm{i}\). Therefore, \(A\) has the eigenvalues \(-\mathrm{i}\) and \(\mathrm{i}\).